Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 4 (2008), 070, 21 pages
The PBW Filtration, Demazure Modules
and Toroidal Current Algebras?
Evgeny FEIGIN
†‡
†
Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931, Cologne, Germany
‡
I.E. Tamm Department of Theoretical Physics, Lebedev Physics Institute,
Leninski Prospect 53, Moscow, 119991, Russia
E-mail: evgfeig@gmail.com
Received July 04, 2008, in final form October 06, 2008; Published online October 14, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/070/
Abstract. Let L be the basic (level one vacuum) representation of the affine Kac–Moody
Lie algebra b
g. The m-th space Fm of the PBW filtration on L is a linear span of vectors
of the form x1 · · · xl v0 , where l ≤ m, xi ∈ b
g and v0 is a highest weight vector of L. In this
paper we give two descriptions of the associated graded space Lgr with respect to the PBW
filtration. The “top-down” description deals with a structure of Lgr as a representation
of the abelianized algebra of generating operators. We prove that the ideal of relations is
generated by the coefficients of the squared field eθ (z)2 , which corresponds to the longest
root θ. The “bottom-up” description deals with the structure of Lgr as a representation
of the current algebra g ⊗ C[t]. We prove that each quotient Fm /Fm−1 can be filtered by
graded deformations of the tensor products of m copies of g.
Key words: affine Kac–Moody algebras; integrable representations; Demazure modules
2000 Mathematics Subject Classification: 17B67
1
Introduction
Let g be a finite-dimensional simple Lie algebra, b
g be the corresponding affine Kac–Moody Lie
algebra (see [21, 25]). Let L be the basic representation of b
g, i.e. an irreducible level one module
with a highest weight vector v0 satisfying condition (g ⊗ C[t]) · v0 = 0. The PBW filtration F•
on the space L is defined as follows:
F0 = Cv0 , Fm+1 = Fm + span{x · w : x ∈ b
g, w ∈ Fm }.
This filtration was introduced in [13] as a tool of study of vertex operators acting on the space
of Virasoro minimal models (see [8]). In this paper we study the associated graded space
Lgr = F0 ⊕ F1 /F0 ⊕ · · · . We describe the space Lgr from two different points of view: via
“top-down” and “bottom-up” operators (the terminology of [24]).
On one hand, the space Lgr is a module over the Abelian Lie algebra gab ⊗t−1 C[t−1 ], where gab
is an Abelian Lie algebra whose underlying vector space is g. The module structure is induced
from the action of the algebra of generating “top-down” operators g ⊗ t−1 C[t−1 ] on L. Thus Lgr
can be identified with a polynomial ring on the space gab ⊗ t−1 C[t−1 ] modulo certain ideal. Our
first goal is to describe this ideal explicitly.
On the other hand, all spaces Fm are stable with respect to the action of the subalgebra of
annihilating operators g⊗C[t] (the “bottom-up” operators). This gives g⊗C[t] module structure
?
This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection
is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html
2
E. Feigin
on each quotient Fm /Fm−1 . Our second goal is to study these modules. We briefly formulate
our results below.
P
For x ∈ g let x(z) = i<0 (x ⊗ ti )z −i be a generating function of the elements x ⊗ ti ∈ b
g,
i < 0. These series are also called fields. They play a crucial role in the theory of vertex operator
algebras (see [15, 22, 3]). We will need the field which corresponds to the highest root θ of g.
Namely, let eθ ∈ g be a highest weight vector in the adjoint representation. It is well known
(see for instance [3]) that the coefficients of eθ (z)2 vanish on L. It follows immediately that the
same relation holds on Lgr . We note also that the Lie algebra g ' g ⊗ 1 acts naturally on Lgr .
The following theorem is one of the central results of our paper.
Theorem 1. Lgr is isomorphic to the quotient of the universal enveloping algebra U(gab ⊗
t−1 C[t−1 ]) by the ideal I, which is the minimal g ⊗ 1 invariant ideal containing all coefficients
of the series eθ (z)2 .
This proves the level one case of the conjecture from [10]. We note that for g = sl2 this
theorem was proved in [13]. The generalization of this theorem for higher levels and g = sl2 is
conjectured in [10].
In order to prove this theorem and to make a connection with the “bottom-up” description
we study the intersection of the PBW filtration with certain Demazure modules inside L. Recall
(see [7]) that by definition a Demazure module D(λ) ,→ L is generated by extremal vector of
the weight λ with an action of the universal enveloping algebra of the Borel subalgebra of b
g.
We will only need the Demazure modules D(N θ). These modules are invariant with respect
to the current algebra g ⊗ C[t] and provide a filtration on L by finite-dimensional spaces:
D(θ) ,→ D(2θ) ,→ · · · = L (see [18, 19]; some special cases are also contained in [12, 23]). Let
Fm (N ) = D(N θ) ∩ Fm be an intersection of the Demazure module with the m-th space of the
PBW filtration. This gives a filtration on D(N θ). In order to describe the filtration F• (N ) we
use a notion of the fusion product of g ⊗ C[t] modules (see [17, 11]) and the Fourier–Littelmann
results [19].
We recall that there exist two versions of the fusion procedure for modules over the current
algebras. The first version constructs a graded g⊗C[t] module V1 ∗· · ·∗VN starting from the tensor
product of cyclic g ⊗ C[t] modules Vi . The other version also produces a graded g ⊗ C[t] module
V1 ∗∗ · · ·∗∗VN , but in this case all Vi are cyclic g modules. (We note that second version is a special
case of the first one). The fusion modules provide a useful tool for the study of the representation
theory of current and affine algebras (see [1, 2, 4, 16, 2, 23, 9, 10, 14, 19]). In particular,
Fourier and Littelmann proved that there exists an isomorphism of g ⊗ C[t] modules D(N θ) '
D(θ) ∗ · · · ∗ D(θ) (N times). Using this theorem and the ∗∗-version of the fusion procedure, we
endow the space D(N θ) with a structure of the representation of the toroidal current algebra
g ⊗ C[t, u] (see [20, 26] and references therein for some details on the representation theory of
the toroidal algebras). This allows to prove our second main theorem:
Theorem 2. The g ⊗ C[t] module Fm (N )/Fm−1 (N ) can be filtered by N
m copies of the
m-th
fusionm power of the adjoint representation of g. In particular, dim Fm (N )/Fm−1 (N ) =
N
m (dim g) .
The paper is organized as follows. In Section 2 we give the definition of the PBW filtration
and of the induced filtration on Demazure modules. In Section 3 we study tensor products
of cyclic g ⊗ C[t] modules endowed with a structure of representations of toroidal algebra.
In particular, we show that fusion product D(1)∗∗N is well defined. In Section 4 the results
of Section 3 are applied to the module D(N ). We prove a graded version of the inequality
m
dim Fm (N )/Fm−1 (N ) ≥ N
m (dim g) . In Section 5 the functional realization of the dual space
(Lgr )∗ is given. In Section 6 we combine all results of the previous sections and prove Theorems 1
and 2. We finish the paper with a list of the main notations.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
2
3
The PBW f iltration
In this section we recall the definition and basic properties of the PBW filtration (see [13]).
Let g be a simple Lie algebra, b
g be the corresponding affine Kac–Moody Lie algebra:
b
g = g ⊗ C[t, t−1 ] ⊕ CK ⊕ Cd.
Here K is a central element, d is a degree element ([d, x ⊗ ti ] = −ix ⊗ ti ) and
[x ⊗ ti , y ⊗ tj ] = [x, y] ⊗ ti+j + iδi+j,0 (x, y)K,
x, y ∈ g, (·, ·) is a Killing form. Let L be the basic representation of the affine Lie algebra, i.e.
level one highest weight irreducible module with a highest weight vector v0 satisfying
(g ⊗ C[t]) · v0 = 0,
Kv0 = v0 ,
dv0 = 0,
−1
−1
U g ⊗ t C[t ] · v0 = L.
The operator d ∈ b
g defines a graded character of any subspace V ,→ L by the formula
chq V =
X
q n dim{v ∈ V : dv = nv}.
n≥0
P
For x ∈ g we introduce a generating function x(z) = i>0 (x ⊗ t−i )z i of the elements x ⊗ ti ,
i < 0. We will mainly deal with the function eθ (z), where θ is the highest weight of g and eθ ∈ g
is a highest weight element. All coefficients
X
(eθ ⊗ ti )(eθ ⊗ tj )
i+j=n
i,j≤−1
of the square of the series eθ (z) are known to vanish on L (this follows from the vertex operator
realization of L [15, Theorem A]). Equivalently, eθ (z)2 = 0 on L.
c2 module into L. Namely,
In what follows we will need a certain embedding of the basic sl
θ
let sl2 be a Lie algebra spanned by a sl2 -triple eθ , fθ and hθ , where eθ and fθ are highest
and lowest weight vectors in the adjoint representation of g. Then the restriction map defines
cθ module on L. In particular, the space (sl
c
a structure of sl
U θ2 ) · v0 is isomorphic to the basic
2
c2 , since the defining relations (eθ ⊗ t−1 )2 v0 = 0 and (slθ ⊗ 1)v0 = 0 are
representation of sl
2
satisfied (see [25, Lemma 2.1.7]).
We now define the PBW filtration F• on L. Namely, let
Fm+1 = Fm + span{(x ⊗ t−i )w, x ∈ g, i > 0, w ∈ Fm }.
F0 = Cv0 ,
Then F• is an increasing filtration converging to L. We denote the associated graded space
by Lgr :
Lgr =
M
Lgr
m,
Lgr
m = grm F• = Fm /Fm−1 .
m≥0
In what follows we denote by gab an Abelian Lie algebra with the underlying vector space
isomorphic to g. We endow gab with a structure of g module via the adjoint action of g on itself.
Lemma 1. The action of g ⊗ C[t] on L induces an action of the same algebra on Lgr . The
action of g ⊗ t−1 C[t−1 ] on L induces an action of the algebra gab ⊗ t−1 C[t−1 ] on Lgr .
4
E. Feigin
Proof . All spaces Fm are invariant with respect to the action of g ⊗ C[t], since the condition
(g ⊗ C[t]) · v0 = 0 is satisfied. Hence we obtain an induced action on the quotient spaces
Fm /Fm−1 .
Operators x ⊗ ti , i < 0 do not preserve Fm but map it to Fm+1 . Therefore, each element
gr
x ⊗ ti , i < 0 produce an operator acting from Lgr
m to Lm+1 . An important property of these
operators on Lgr is that they mutually commute, since the composition (x ⊗ ti )(y ⊗ tj ) acts
from Fm to Fm+2 but the commutator [x ⊗ ti , y ⊗ tj ] = [x, y] ⊗ ti+j maps Fm to Fm+1 . Lemma
is proved.
The goal of our paper is to describe the structure of Lgr as a representation of g ⊗ C[t] and
of gab ⊗ t−1 C[t−1 ]. It turns out that these two structures are closely related.
Lemma 2. Let I ,→ U(gab ⊗ t−1 C[t−1 ]) be the minimal g-invariant ideal containing all coefficients of the series eθ (z)2 . Then there exists a surjective homomorphism
ab
−1
−1
gr
U g ⊗ t C[t ] /I → L
(1)
of gab ⊗ t−1 C[t−1 ] modules mapping 1 to v0 .
Proof . Follows from the relation eθ (z)2 v0 = 0.
One of our goals is to prove that the homomorphism (1) is an isomorphism.
Recall (see [18]) that L is filtered by finite-dimensional Demazure modules [7]. A Demazure
module D is generated by an extremal vector with the action of the algebra of generating
operators. We will only need special class of Demazure modules. Namely, for N ≥ 0 let vN ∈ L
be the vector of weight N θ defined by
vN = eθ ⊗ t−N
N
v0 .
We recall that N θ is an extremal weight for L and thus vN spans weight N θ subspace of L.
Let D(N ) ,→ L be the Demazure module generated by vector vN . Thus D(N ) is cyclic g ⊗ C[t]
module with cyclic vector vN . It is known (see [25, 18]) that these modules are embedded into
each other and the limit coincides with L:
D(1) ,→ D(2) ,→ · · · = L.
We introduce an induced PBW filtration on D(N ):
F0 (N ) ,→ F1 (N ) ,→ · · · = D(N ),
Fm (N ) = D(N ) ∩ Fm .
(2)
Obviously, each space Fm (N ) is g ⊗ C[t] invariant.
Lemma 3. FN (N ) = D(N ), but FN −1 (N ) 6= D(N ).
Proof . First equality holds since (eθ ⊗t−N )N v0 = vN and D(N ) is cyclic. In addition vN ∈
/ FN −1
because all weights of Fm (as a representation of g ' g ⊗ 1) are less than or equal to mθ.
Consider the associated graded object
grF (N ) =
∞
M
grm F (N ),
grm F (N ) = Fm (N )/Fm−1 (N ).
m=0
We note that each space grm F (N ) has a natural structure of g ⊗ C[t] module.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
3
5
tN -f iltration
In this section we describe the filtration (2) using the generalization of the fusion product of
g ⊗ C[t] modules from [17] and a theorem of [19]. We first recall the definition of the fusion
product of g ⊗ C[t] modules.
Let V be a g⊗C[t] module, c be a complex number. We denote by V (c) a g⊗C[t] module which
coincides with V as a vector space and the action is twisted by the Lie algebra homomorphism
φ(c) : g ⊗ C[t] → g ⊗ C[t],
x ⊗ tk 7→ x ⊗ (t + c)k .
Let V1 , . . . , VN be cyclic representations of the current algebra g ⊗ C[t] with cyclic vectors
v1 , . . . , vN . Let c1 , . . . , cN be a set of pairwise distinct complex numbers. The fusion product
V1 (c1 ) ∗ · · · ∗ VN (cN ) is a graded deformation of the usual tensor product V1 (c1 ) ⊗ · · · ⊗ VN (cN ).
More precisely, let U(g ⊗ C[t])s be a natural grading on the universal enveloping algebra coming
from the counting of the t degree. Because of the condition ci 6= cj for i 6= j, the tensor product
NN
N
i=1 Vi (ci ) is a cyclic U(g ⊗ C[t]) module with a cyclic vector ⊗i=1 vi . Therefore, the grading
on U(g ⊗ C[t]) induces an increasing fusion filtration
U(g ⊗ C[t])≤s · (v1 ⊗ · · · ⊗ vN )
(3)
on the tensor product.
Definition 1. The fusion product V1 (c1 ) ∗ · · · ∗ VN (cN ) of g ⊗ C[t] modules Vi is an associated
graded g ⊗ C[t] module with respect to the filtration (3) on the tensor product V1 (c1 ) ⊗ · · · ⊗
VN (cN ). We denote the m-th graded component by grm (V1 (c1 ) ∗ · · · ∗ VN (cN )).
We note that in many special cases the g ⊗ C[t] module structure of the fusion product does
not depend on the parameters ci (see for example [1, 5, 11, 19, 16]). We then omit the parameters
ci and denote the corresponding fusion product simply by V1 ∗ · · · ∗ VN .
In what follows we will need a special but important case of the procedure described above.
Namely, let Vi be cyclic g modules. One can extend the g module structure to the g ⊗ C[t]
module structure by letting g ⊗ tC[t] to act by zero. We denote the corresponding g ⊗ C[t]
modules by V i .
Definition 2. Let V1 , . . . , VN be a set of cyclic g modules. Then a g ⊗ C[t] module V1 (c1 ) ∗
∗ · · · ∗ ∗VN (cN ) is defined by the formula:
V1 (c1 ) ∗ ∗ · · · ∗ ∗VN (cN ) = V 1 (c1 ) ∗ · · · ∗ V N (cN ).
We denote the m-th graded component by grm (V1 (c1 ) ∗ ∗ · · · ∗ ∗VN (cN )).
Remark 1. The fusion procedure described in Definition 2 can be reformulated as follows. One
starts with a tensor product of evaluation g ⊗ C[t] modules Vi (ci ), where x ⊗ tk acts on Vi by cki x
(we evaluate t at the point ci ). Then one constructs the fusion filtration and associated graded
module (according to Definition 1).
Remark 2. Let us stress the main difference between Definitions 1 and 2. Definition 1 gets as
an input a set of cyclic representations of the current algebra g ⊗ C[t] and as a result produces
a t-graded g ⊗ C[t] module. The input of Definition 2 is a set of cyclic g modules and an output
is again a t-graded g ⊗ C[t] module.
We now recall a theorem from [19] which uses the fusion procedure to construct the Demazure
module D(N ) starting from D(1).
6
E. Feigin
Theorem 3 ([19]). The N -th fusion power D(1)∗N is independent on the evaluation parameters ci . The g ⊗ C[t] modules D(1)∗N and D(N ) are isomorphic.
We recall that as a g module D(1) is isomorphic to the direct sum of trivial and adjoint
representations, D(1) ' C ⊕ g. The trivial representation is annihilated by g ⊗ C[t], the adjoint
representation is annihilated by g ⊗ t2 C[t] and g ⊗ C[t] maps g to C.
Our idea is to combine the theorem above and Definition 2 with g being the current algebra
g ⊗ C[u] (see also [4], where Definition 2 is used in affine settings). Definition 2 works for
arbitrary g and produces a representation of an algebra with an additional current variable. In
particular, starting from the g ⊗ C[u] modules Vi = D(1) and an N tuple of pairwise distinct
complex numbers c1 , . . . , cN one gets a new bi-graded g ⊗ C[t, u] module. The resulting module
can be obtained from the Demazure module D(N ) by a rather simple procedure which we are
going to describe now.
Let V1 , . . . , VN be cyclic representations of the algebra g⊗C[t]/ht2 i. Hence V1 (c1 )∗· · ·∗VN (cN )
is a cyclic g⊗C[t]/ht2N i module. We consider a decreasing filtration U(g⊗C[t])j on the universal
enveloping algebra defined by
0
j+1
= g ⊗ tN C[t] U(g ⊗ C[t])j .
(4)
U(g ⊗ C[t]) = U(g ⊗ C[t]),
U(g ⊗ C[t])
This filtration induces a decreasing filtration Gj on the fusion product V1 (c1 ) ∗ · · · ∗ VN (cN )
(since it is cyclic U(g ⊗ C[t]) module). G• will be also referred to as a tN -filtration. Consider
the associated graded space
•
grG =
N
M
grj G• ,
grj G• = Gj /Gj+1 .
(5)
j=0
Since each space Gj is g ⊗ C[t] invariant one gets a structure of g ⊗ C[t]/htN i module on each
space Gj /Gj+1 . In addition an element from g ⊗ tN C[t]/ht2N i produces a degree 1 operator
on (5) mapping grj G• to grj+1 G• . We thus obtain a structure of g ⊗ C[t, u]/htN , u2 i module
on (5), where g ⊗ uC[t] denotes an algebra of degree one operators on grG• coming from the
action of g ⊗ tN C[t]/ht2N i.
On the other hand let us consider the modules Vi as representations of the Lie algebra
g ⊗ C[u]/hu2 i (simply replacing t by u). We denote these modules as Viu . Then the bi-graded
tensor product V1u (c1 ) ∗ ∗ · · · ∗ ∗VNu (cN ) is a representation of the Lie algebra g ⊗ C[t, u]/htN , u2 i.
Proposition 1. We have an isomorphism of g ⊗ C[t, u]/htN , u2 i modules
grG• ' V1u (c1 ) ∗ ∗ · · · ∗ ∗VNu (cN ).
(6)
Proof . The idea of the proof is as follows. We start with the tensor product V1 ⊗ · · · ⊗ VN
and apply the fusion filtration. Afterwards we apply the tN -filtration G• . Combining these
operations with certain changes of basis of current algebra we arrive at the definition of the
bi-graded module V1u ∗ ∗ · · · ∗ ∗VNu . We give details below.
For an element x⊗ti ∈ g⊗C[t] let (x⊗ti )(j) be the operator on the tensor product V1 ⊗· · ·⊗VN
defined by
(x ⊗ ti )(j) = Id⊗j−1 ⊗ (x ⊗ ti ) ⊗ Id⊗N −j ,
i.e. (x ⊗ ti )(j) acts as x ⊗ ti on Vj and as an identity operator on the other factors. In order to
construct the fusion product one starts with the operators
A(x ⊗ ti ) =
N
X
j=1
x ⊗ (t + cj )i
(j)
,
(7)
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
7
where cj are pairwise distinct numbers. These operators define an action of the algebra g ⊗ C[t]
on the tensor product V1 (c1 ) ⊗ · · · ⊗ VN (cN ). Since x ⊗ ti with i > 1 vanish on Vj we obtain
A(x ⊗ ti ) =
N
X
N
X
cij (x ⊗ 1)(j) +
j=1
icji−1 (x ⊗ t)(j) .
j=1
The next step is to pass to the associated graded module with respect to the fusion filtration.
By definition, operators of the form
A x ⊗ tk + linear combination of A x ⊗ tl ,
l<k
(8)
act on the associated graded module in the same way as A(x ⊗ tk ) (the lower degree term vanish
after passing to the associated graded space). We are going to fix special linear changes in (8)
for N ≤ k < 2N which make the expressions for A(x ⊗ tk ) simpler.
Let α0 , α1 , . . . , αN −1 be numbers such that for all 1 ≤ j ≤ N
cN
j +
N
−1
X
αi cij = 0.
i=0
We state that
A x⊗t
N +s
+
N
−1
X
αi A x ⊗ t
s+i
=
i=0
N
X
csj (x ⊗ t)(j)
j=1
Y
(cj − ck )
(9)
k6=j
for all 0 ≤ s ≤ N − 1. Let
f (x) = xN + αN −1 xN −1 + · · · + α0 .
Then f (x) =
QN
− ck ). Therefore, for the derivative csj f 0 (cj ) one gets
k=1 (x
+s−1
csj f 0 (cj ) = N cN
+
j
N
−1
X
iαi ci+s−1
= csj
j
i=0
Y
(cj − ck ).
k6=j
This proves (9).
Using formula (9), we replace operators A(x ⊗ ti ), 0 ≤ i < 2N by operators B(x ⊗ ti ) as
follows
N
N
X
X
(j)
B x ⊗ ti =
cij (x ⊗ 1)(j) +
ici−1
j (x ⊗ t) ,
j=1
B x ⊗ tN +i =
N
X
i=1
0 ≤ i < N,
(10)
j=1
csi (x ⊗ t)(i)
Y
(cj − ck ),
0 ≤ i < N,
k6=j
thus performing the linear change (8). So we redefine half of the operators A(x ⊗ ti ) and leave
the other half unchanged.
In order to construct the left hand side of (6) one first applies the fusion filtration to the
algebra of operators B(x ⊗ ti ) and afterwards proceeds with the tN -filtration. The last step
means that the subtraction of a linear combination of the operators B(x ⊗ tN +i ), 0 ≤ i < N
8
E. Feigin
from B(x ⊗ ti ), 0 ≤ i < N does not change the structure of the resulting module. Redefining
the operators (10) we arrive at the following operators:
C x⊗t
i
=
N
X
cij (x ⊗ 1)(j) ,
0 ≤ i < N,
j=1
C x⊗t
N +i
=
N
X
cij (x ⊗ t)(j) ,
0 ≤ i < N.
i=j
Q
Note that we can remove constants k6=j (cj − ck ) from B(x ⊗ tN +i ) since this procedure corresponds simply to rescaling the variable t in each Vi .
Summarizing all the formulas above we arrive at the following two steps construction of the
left hand side of (6):
• apply the fusion procedure to the operators C(x ⊗ ti ), 0 ≤ i < 2N ,
• attach a u-degree 1 to each of the operators C(x ⊗ tN +i ), 0 ≤ i < 0.
In order to construct the right hand side of (6) one uses the same procedure with C(x ⊗ tN +i )
being operators which correspond to x ⊗ uti (see (4) and (5)). Thus we have shown that the
associated graded to the fusion product with respect to the tN -filtration is isomorphic to the
module V1 ∗ ∗ · · · ∗ ∗VN . Proposition is proved.
Corollary 1. The fusion product D(1) ∗ ∗ · · · ∗ ∗D(1) does not depend on the evaluation parameters.
Corollary 2. The fusion product of the adjoint representations
g(c1 ) ∗ ∗ · · · ∗ ∗g(cN )
is independent of the parameters c1 , . . . , cN .
Proof . By definition, the zeroth graded component with respect to the t grading of the module
D(1)(c1 ) ∗ ∗ · · · ∗ ∗D(1)(cN )
is isomorphic to the fusion product g(c1 ) ∗ ∗ · · · ∗ ∗g(cN ). From the Proposition above we obtain
an isomorphism
g(c1 ) ∗ ∗ · · · ∗ ∗g(cn ) ' D(N )/(g ⊗ tN C[t])D(N )
for any c1 , . . . , cn . Thus the left hand side is independent of ci .
We finish this section introducing an “energy” operator d¯ on the fusion product. The operator d¯ acts by a constant m on the graded component
grm (V1 (c1 ) ∗ · · · ∗ VN (cN )).
The operator d¯ defines a graded character of V1 (c1 )∗· · ·∗Vn (cn ) by the standard formula formula
X
¯ = nv}.
chq V1 (c1 ) ∗ · · · ∗ Vn (cn ) =
q n dim{v : dv
n≥0
An analogous formula defines a character of V1 (c1 ) ∗ ∗ · · · ∗ ∗Vn (cn ). We note that this grading
has nothing to do with the Cartan grading coming from the action of the Cartan subalgebra.
We do not consider the latter grading in this paper.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
9
Remark 3. Let Vi ' D(1) for all i. Then the fusion module is independent on the evaluation
parameters and D(1)∗N is isomorphic to the Demazure module D(N ) (see [19]). By the very
definition we have an embedding D(N ) ,→ L. Thus both operators d and d¯ are acting on D(N ),
satisfying the relations
[d, x ⊗ ti ] = −ix ⊗ ti ,
¯ x ⊗ ti ] = ix ⊗ ti .
[d,
¯ N = 0 we have a simple identity d¯ = N 2 − d.
Since dvN = N 2 vN and dv
4
Demazure modules
In this section we study the fusion filtration on the tensor product D(1)⊗N and the induced
PBW filtration on the Demazure modules D(N ). We also derive some connections between
these filtrations.
Let Du (1) be the g ⊗ C[u]/hu2 i module obtained from D(1) by substituting u instead of t.
In particular, (Du (1))∗∗N is a (t, u) bi-graded representation of the algebra g ⊗ C[t, u]/htN , u2 i.
Here t-grading is exactly the fusion grading grj Du (1) ∗ · · · ∗ Du (1) and the u-grading comes from
the grading on Du (1), which assigns degree zero to g and degree one to Cv0 . We consider the
decomposition
(Du (1))∗∗N =
N
M
W (m)
m=0
into the graded components with respect to the u-grading. (The u-grading is bounded from above
by N since the u-grading in each of D(1) could be either 0 or 1). Note that each space W (m) is
a representation of g ⊗ C[t]. We want to show that W (m) can be filtered by certain number of
copies of the fusion product g∗∗N −m . The precise statement is given in the following proposition.
Proposition 2. Let Du (1)∗∗N be a bi-graded tensor product of N copies of g⊗C[u]-module D(1).
Then
• For any 0 ≤ m ≤ N the g ⊗ C[t] module W (m) can be filtered by N
m copies of the
g ⊗ C[t]-module g∗∗N −m .
• The cyclic vectors of the modules g∗∗N −m above are the images of the vectors
f θ ⊗ uti1 · · · f θ ⊗ utim vN ,
0 ≤ i1 ≤ · · · ≤ im ≤ N − m.
(11)
Remark 4. We first give a non rigorous, but conceptual explanation of the statement of the
proposition above. Recall that Du (1) is isomorphic to g ⊕ C as a g module. Let v1 , v0 ∈ D(1)
be highest weight vectors of g and C respectively. Then (fθ ⊗ u)v1 = v0 and (fθ ⊗ u)2 v0 = 0.
Therefore, after making the fusion Du (1)∗∗N , the tensor product of N copies of 2-dimensional
vector space span{v0 , v1 } will be deformed into the N -fold fusion product of two-dimensional
representations of the algebra C[fθ ]. The set of vectors (11) represents a basis of this fusion
product (see [6, 11]). Hence Du (1)∗N is equal to the U(g ⊗ C[t]) span of the vectors of the
form (11). We now want to describe the space U(g ⊗ C[t]) · wm , where wm is of the form (11)
with exactly m factors. Note that wm is a linear combination of the vectors of the form
vi1 ⊗ · · · ⊗ viN ,
where iα equal 0 or 1 and the number of α such that iα = 0 is equal to m. This means that the
⊗m .
space U(g ⊗ C[t]) · wm is embedded into the direct sum of N
m copies of the tensor product g
10
E. Feigin
Hence it is natural to expect that after passing to the associated graded object with respect to
the fusion filtration one arrives at the fusion product g∗∗m . This is not exactly true. In order
to make the statement precise one additional filtration is needed (that is the reason why W (m)
is not the direct sum of the fusion products, but rather can be filtered by these modules).
We now give the proof of Proposition 2.
Proof . As a starting point we note the isomorphism of g ⊗ C[t]-modules
W (0) ' g∗∗N .
In fact, D(1) ' g ⊕ Cv0 with cyclic vector being the highest weight vector of g. The algebra
g ⊗ u maps C to zero and g to C. In particular,
(fθ ⊗ u) · v1 = v0 .
Hence if we do not apply operators with positive powers of u (i.e. we consider the space W (0))
we arrive at the usual fusion product of N copies of g.
We now introduce a decreasing filtration W j (m) such that the associated graded object is
∗∗m . Let
isomorphic to the direct sum of N
m copies of g
wi1 ,...,im = f θ ⊗ uti1 . . . f θ ⊗ utim vN ,
0 ≤ i1 ≤ · · · ≤ im ≤ N − m.
We set
W n (m) = U(g ⊗ C[t]) · span{wi1 ,...,im : i1 + · · · + im ≥ n}.
In particular, W 0 (m) = W (m) and each space W n (m) is g ⊗ C[t] invariant. We state that the
associated graded space
W 0 (m)/W 1 (m) ⊕ W 1 (m)/W 2 (m) ⊕ · · ·
(12)
∗∗N −m . Moreover the highest
is isomorphic to the direct sum of N
m copies of the modules g
weight vectors of these modules are exactly the images of wi1 ,...,im . We prove this statement for
m = 1. The proof for other m is very similar.
For m = 1 we have
i
wi = fθ ⊗ ut =
N
X
cij v1⊗j−1 ⊗ v0 ⊗ v1⊗N −j ,
i = 0, . . . , N − 1.
j=1
We want to show that
W i (1)/W i+1 (1) ' g∗∗N −1 ,
W i (1) = U(g ⊗ C[t]) · span{wi , . . . , wN −1 }.
Let αi,j , 1 ≤ i, j ≤ N be some numbers. Denote


N
N
X

X
i
W̃ (1) = U(g ⊗ C[t]) · span
αi+1,j wi , . . . ,
αN,j wN −1 .


j=1
j=1
We state that
W i (1)/W i+1 (1) ' W̃ i (1)/W̃ i+1 (1)
(13)
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
11
as g ⊗ C[t] modules. In fact, adding to wi a linear combination of wj with j < i does not change
the g ⊗ C[t] module structure because of the fusion filtration. Because of the filtration (12), this
is still true if one adds a linear combination of wj with j > i. Thus we conclude that we can
replace each vector wi by an arbitrary linear combination. In particular, there exist numbers αi,j
such that
N
X
αi−1,j wi = v0⊗i ⊗ v1 ⊗ v0⊗N −1−i .
j=1
Since v1 is a highest weight vector of the adjoint representation of g and v0 spans the trivial
representation, we arrive at the fact that
W̃ i (1)/W̃ i+1 (1) ' g∗∗N −1 .
Because of the isomorphism (13), the m = 1 case of the proposition is proved.
We are now going to connect the filtrations G• (N ) and the induced PBW filtration F• (N )
on the Demazure modules D(N ). We use Proposition 1 for Vi = D(1).
Lemma 4. Gm is a subspace of FN −m (N ).
Proof . We first note that for the cyclic vector vN ∈ D(N ) we have
vN = eθ ⊗ t−N
N
v0 .
Therefore, G0 = FN (N ) = D(N ) and our Lemma is true for m = 0.
In general, we need to prove that
x1 ⊗ tN +i1 · · · xm ⊗ tN +im v ∈ FN −m
(14)
(15)
for any v ∈ D(N ) and x1 , . . . , xm ∈ g, i1 , . . . , im ≥ 0. Since v ∈ D(N ) there exists an element
r ∈ U(g ⊗ C[t]) such that v = rvN . Because of (14), the inclusion (15) follows from the following
statement:
N
x1 ⊗ tN +i1 · · · xm ⊗ tN +im y1 ⊗ tj1 · · · ym ⊗ tjm eθ ⊗ t−N v0 ∈ FN −m
(16)
for arbitrary xα , yβ ∈ g and iα , jβ , n ≥ 0. Since (g ⊗ C[t]) · v0 = 0 the expression
N
y1 ⊗ tj1 · · · yn ⊗ tjn eθ ⊗ t−N v0
is equal to a linear combination of the vectors of the form
z1 ⊗ t−N +l1 · · · zN ⊗ t−N +lN v0
where zi ∈ g and li ≥ 0. One can easily see that because of the condition (g ⊗ C[t])v0 = 0 the
expression of the form
x ⊗ tN +i z1 ⊗ t−N +l1 · · · zN ⊗ t−N +lN v0
can be rewritten as a linear combination of the monomials of the form
z1 ⊗ t−N +l1 · · · zN −1 ⊗ t−N +1+lN −1 v0 .
Iterating this procedure we arrive at (16). This proves (15) and hence our lemma is proved. 12
E. Feigin
Recall a notation for q-binomial coefficients
n
(q)n
=
,
(q)a = (1 − q) · · · (1 − q a ).
m q
(q)m (q)n−m
P
P
For two q series f (q) = fn q n and g(q) = gn q n we write f ≥ g if fn ≥ gn for all n.
Corollary 3. The following character inequality holds:
m2 N
chq grm F• (N ) ≥ q
ch −1 g∗∗m .
m q q
(17)
Proof . Because of the Lemma above, it suffices to show that
m
m+1
(N −m)2 N
chq G /G
=q
ch −1 g∗∗N −m .
m q q
Note that dvN = N 2 vN . Therefore, the graded character of the space of cyclic vectors wi1 ,...,im
from Proposition 2 (where
the q-degree of u is fixed to be equal to N according to ProposiN
2
. Multiplying by the d¯ character of g∗∗N −m with respect to d¯ (see
tion 1) is given by q m
m q
Remark 3), we arrive at our Corollary.
Corollary 4. The following character inequality holds:
2
chq grm F• ≥ q m
1
ch −1 g∗∗m .
(q)m q
Proof . Follows from limit formulas
N
1
.
lim D(N ) = L,
lim
=
N →∞
N →∞ m q
(q)m
(18)
Remark 5. In the next section we prove that (18) is an equality. In Section 6 we prove that (17)
is also an equality.
5
Dual functional realization
We now consider the restricted dual space to the “expected” PBW filtered space Lgr . Let
U
ab
= C[gab ⊗ t−1 C[t−1 ]]
be a space of polynomial functions on the space gab ⊗ t−1 C[t−1 ] (recall that gab is an Abelian Lie
algebra with the underlying vector space isomorphic to g). The algebra Uab is an abelinization
of the universal enveloping algebra U(g ⊗ t−1 C[t−1 ]) of the Lie algebra of generating operators
(due to the PBW theorem). We note that g acts on the space gab ⊗ t−1 C[t−1 ] via the adjoint
representation on gab .
Let I ∈ Uab be the minimal g invariant ideal, which contains all coefficients of eθ (z)2 , i.e. all
elements of the form
X
eθ ⊗ t−i eθ ⊗ t−n−1+i ,
n = 1, 2, . . . .
1≤i≤n
Denote
L0 = Uab /I.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
13
This space can be decomposed according to the number of variables in a monomial:
0
L =
∞
M
L0m
=
m=0
∞
M
span x1 ⊗ ti1 · · · xm ⊗ tim , xi ∈ gab .
m=0
The operator d ∈ b
g induces a degree operator on L0 . We denote this operator by the same
symbol. There exists a surjective homomorphism of gab ⊗ t−1 C[t−1 ] modules L0 → Lgr (see
Lemma 2). Our goal is to show that L0 ' Lgr . Let (L0m )∗ be a restricted dual space:
M
(L0m,n )∗ ,
L0m,n = {v ∈ L0m : dv = nv}.
(L0m )∗ =
n≥0
We construct the functional realization of L0m using currents
X
x(z) =
x ⊗ t−i z i
i>0
for x ∈ g. Following [13] we consider a map
αm : (L0m )∗ → C[z1 , . . . , zm ] ⊗ g⊗m ,
φ 7→ rφ
from the dual space (L0m )∗ to the space of polynomials in m variables with values in the m-th
tensor power of the space g. This map is given by the formula
hrφ , x1 ⊗ · · · ⊗ xm i = φ(x1 (z1 ) · · · xm (zm )),
where brackets in the left hand side denote the product of non degenerate Killing forms on n
factors g. Our goal is to describe the image of αm . We first formulate the conditions on rφ ,
which follow from the definition of L0 and then prove that these conditions determine the image
of αm . We prepare some notations first.
Consider the decomposition of the tensor square g ⊗ g into the direct sum of g modules:
g ⊗ g = V2θ ⊕
^2
g ⊕ S 2 g/V2θ .
(19)
Here V2θ is a highest weight g-module with a highest weight 2θ embedded into S 2 g via the map
V2θ ' U(g) · (eθ ⊗ eθ ) ,→ S 2 g.
Lemma 5. For the module g ∗ ∗g we have
gr0 (g ∗ ∗g) = V2θ ,
gr1 (g ∗ ∗g) =
^2
g,
gr2 (g ∗ ∗g) = S 2 g/V2θ
and all other graded components vanish.
Proof . We first show that grn (g ∗ ∗g) = 0 for n > 2. Let c1 , c2 be evaluation constants which
appear in Definition 2 of the fusion product. Recall that g ∗ ∗g is independent of the evaluation
parameters. So we can set c1 = 1, c2 = 0. Then the second space of the fusion filtration is
given by
U(g) · (span{[x1 , [x2 , eθ ]], x1 , x2 ∈ g} ⊗ eθ ) ,
(20)
where U(g) acts on the tensor product g ⊗ g diagonally. But [fθ , [fθ , eθ ]] = −2fθ and hence
span{[x1 , [x2 , eθ ]], x1 , x2 ∈ g} = g
14
E. Feigin
(since the left hand side is invariant with respect to the subalgebra of g of annihilating operators
and contains the lowest weight vector fθ of the adjoint representation). Therefore, (20) coincides
with g ⊗ g.
We now compute three nontrivial graded components gr0 (g ∗ ∗g), gr1 (g ∗ ∗g) and gr2 (g ∗ ∗g).
From the definition of the fusion filtration we have
gr0 (g ∗ ∗g) = U(g) · (eθ ⊗ eθ ) ' V (2θ).
We now redefine the evaluation parameters by setting c1 = 1, c2 = −1. Then the formulaVfor
the operators x ⊗ t acting on g ⊗ g is given by x ⊗ Id − Id ⊗ x. These operators map S 2 (g) to 2 g
and vice versa.We conclude that
^2
g,
gr2 (g ∗ ∗g) ,→ S 2 (g).
gr1 (g ∗ ∗g) ,→
Now our Lemma follows from the equality grn (g ∗ ∗g) = 0 for n > 2.
Lemma 6. For any φ ∈ (L0m )∗ the image rφ is divisible by the product z1 · · · zm and satisfies
the vanishing condition
hrφ , V2θ ⊗ g⊗m−2 iz1 =z2 = 0
(21)
and the symmetry condition
σ ∈ Sm ,
σr = r,
(22)
where the permutation group Sm acts simultaneously on the set of variables z1 , . . . , zm and on
the tensor product g⊗m .
Proof . The product z1 · · · zm comes from the condition that the highest weight vector is annihilated by g ⊗ C[t], so for any x ∈ g the series x(z) starts with z. The condition (21) follows
from the relation eθ (z)2 = 0 and g-invariance of the ideal I. The symmetry condition follows
from the commutativity of the algebra Uab .
We denote by
Vm ,→ z1 · · · zm C[z1 , . . . , zm ] ⊗ g⊗m
the space of functions satisfying conditions (21) and (22). In the following lemma we endow Vm
with structures of representation of the ring of symmetric polynomials
sym
Pm
= C[z1 , . . . , zm ]Sm
and of the current algebra g ⊗ C[t].
sym
Lemma 7. There exists natural structures of representations of Pm
and of g ⊗ C[t] on Vm
defined by the following rule:
sym
• Pm
acts on Vm by multiplication on the first tensor factor.
• Lie algebra g ⊗ C[t] acts on Vm by the formula
x⊗t
k
acts as
n
X
zik ⊗ x(i) .
i=1
sym
The actions of Pm
and of g ⊗ C[t] commute.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
15
Proof . A direct computation.
sym
Lemma 7 gives a structure of g ⊗ C[t] module on the quotient space Vm /Pm+
Vm , where the
subscript + denotes the space of polynomials of positive degree. We will show that the dual to
this module is isomorphic to g∗∗m . We first consider the m = 2 case.
Lemma 8. We have an isomorphism of representations of g ⊗ C[t]
∗
V2 /P+sym V2 ' g ∗ ∗g.
Proof . Let r be an element of V2 . Using the decomposition (19), we write r = r0 + r1 + r2 ,
where
r0 ∈ z1 z2 C[z1 , z2 ] ⊗ V2θ ,
r1 ∈ z1 z2 C[z1 , z2 ] ⊗ Λ2 g,
r2 ∈ z1 z2 C[z1 , z2 ] ⊗ S 2 g/V2θ .
Then the conditions (21) and (22) are equivalent to
r0 ∈ z1 z2 (z1 − z2 )2 P sym ⊗ V2θ ,
r1 ∈ z1 z2 (z1 − z2 )P sym ⊗ Λ2 g,
r2 ∈ z1 z2 P sym ⊗ S 2 g/V2θ .
Therefore, the dual quotient space (V2 /P+sym V2 )∗ is isomorphic to g ∗ ∗g as a vector space. It is
straightforward to check that the g ⊗ C[t] modules structures are also isomorphic.
sym
We will also need a nonsymmetric version of the space Vm /Pm+
Vm . Namely, let Wm be a
⊗m
subspace of z1 · · · zm C[z1 , . . . , zm ] ⊗ g
satisfying
Fzi =zj ,→ C[z1 , . . . , zm ]|z1 =z2 ⊗ µi,j S 2 g/V2θ ⊗ g⊗m−2 ,
(23)
2
⊗m−2
∂zi Fzi =zj ,→ C[z1 , . . . , zm ]|z1 =z2 ⊗ µi,j S (g)/V2θ ⊗ g
(24)
for all 1 ≤ i < j ≤ m, where µi,j is an operator on the tensor power g⊗m which inserts the first
two factors on the i-th and j-th places respectively:
µi,j (v1 ⊗ v2 ⊗ v2 ⊗ · · · ⊗ vn ) = v3 ⊗ · · · ⊗ vi−1 ⊗ v1 ⊗ vi ⊗ · · · ⊗ vj−2 ⊗ v2 ⊗ vj−1 · · · ⊗ vn .
The natural action of the polynomial ring Pm = C[z1 , . . . , zm ] commutes with the action of the
current algebra, defined as in Lemma 7. Therefore, we obtain a structure of g ⊗ C[t] module
on the quotient space Wm /Pm+ Wm , where Pm+ is the ring of positive degree polynomials. As
proved in [13], Lemma 5.8, the symmetric and nonsymmetric constructions produce the same
module (see Proposition 3 below for the precise statement). We illustrate this in the case n = 2.
Lemma 9. (W2 /P2+ W2 )∗ ' g ∗ ∗g.
Proof . We use the decomposition r = r0 + r1 + r2 as in Lemma 8 for r ∈ Wm . The conditions (23) and (24) are equivalent to the following conditions on ri :
r0 ∈ (z1 − z2 )2 z1 z2 C[z1 , z2 ] ⊗ V2θ ,
r1 ∈ (z1 − z2 )z1 z2 C[z1 , z2 ] ⊗ Λ2 g,
r2 ∈ z1 z2 C[z1 , z2 ] ⊗ S 2 g/V2θ .
After passing to the quotient with respect to the action of the algebra P2+ we arrive at the
isomorphism of vector spaces (W2 /P2+ W2 )∗ ' g ∗ g. It is straightforward to check that this is
an isomorphism of g ⊗ C[t] modules.
16
E. Feigin
The following Proposition is proved in [13, Lemma 5.8] for g = sl2 .
Proposition 3. There exists an isomorphism of g ⊗ C[t]-modules
sym
Vm /Pm+
Vm ' Wm /Pm+ Wm .
Proof . The proof of the general case differs from the one from [13] by the replacement of the
decomposition of the tensor square of the adjoint representation of sl2 by the general decomposition (19).
Remark 6. We note that the definition of the space Wm is a bit more involved than the
definition of Vm . The reason is that the polynomials used to construct Wm are not symmetric.
sym
In particular, this forces to add the condition (24) in order to get the isomorphism Vm /Pm+
Vm '
Wm /Pm+ Wm .
Proposition 4. The module Wm /P+ Wm is cocyclic with a cocyclic vector being the class of
Y
(zi − zj )2 ⊗ (eθ )⊗m .
r m = z1 · · · zm
1≤i<j≤m
Proof . We first show that if r ∈ Wm is a nonzero element satisfying r ∈ Vmθ ⊗C[z1 , . . . , zm ] then
either r ∈ Pm+ Wm or there exists an element of the universal enveloping algebra of g⊗C[t] which
sends r to rm (here we embed Vmθ into g⊗m as an irreducible component containing (eθ )⊗m ).
In fact, from conditions (23),
Q (24) and the assumption r ∈ Vmθ ⊗ C[z1 , . . . , zm ] we obtain that r
is divisible by the product 1≤i<j≤m (zi − zj )2 . If r ∈
/ Pm+ Wm then we obtain
Y
r =x⊗
(zi − zj )2
1≤i<j≤m
with some x ∈ Vmθ . Since Vmθ is irreducible the U(g) orbit of x contains (eθ )⊗m .
Thus it suffices to show that any element r ∈ Wm /P+ Wm is contained in the U(g ⊗ C[t])
orbit of the image of Vmθ ⊗ C[z1 , . . . , zm ] in the quotient space. We prove that if (a ⊗ t)r = 0
for all a ∈ g then r ∈ Vmθ ⊗ Pm . This would imply the previous statement since Wm /Pm+ Wm
is finite-dimensional.
So let r ∈ Wm be some element and r̄ ∈ Wm /Pm+ Wm be its class. Assume that (a ⊗ t)r = 0
for all a ∈ g. We first show that r ∈ V2θ ⊗ g⊗n−2 ⊗ Pm . In fact, there exists a polynomial l ∈ Pm
such that the following holds in Wm :
(a ⊗ t)r = l(z1 , . . . , zm )r1
1,2
for some r1 ∈ Wm . Consider the space of functions Wm
which satisfy conditions (23) and (24)
1,2
only for i = 1, j = 2. (In particular, Wm ,→ Wm ). We have an isomorphism of g ⊗ C[t] modules
1,2
Wm
' W2 ⊗ g⊗m−2 ⊗ C[z3 , . . . , zm ].
The equality (a ⊗ t)r = lr1 gives
(1)
a
(2)
⊗ z1 + a
⊗ z2 r = lr1 −
m
X
!
(i)
a
⊗ zi
r.
i=3
Using Lemma 9, we obtain r ∈ V2θ ⊗ g⊗n−2 ⊗ Pm .
The same procedure can be done for all pairs 1 ≤ i < j ≤ n. Now our proposition follows
from the following equality in g⊗n :
\
µi,j V2θ ⊗ g⊗n−2 = Vnθ .
1≤i<j≤n
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
17
0 a cyclic vector which corresponds
The dual module (Wm /Pm Wm )∗ is cyclic. We denote by rm
to the cocyclic vector rm .
Proposition 5. There exists a surjective homomorphism of g ⊗ C[t] modules
g∗∗m → (Wm /Pm+ Wm )∗
(25)
0 .
sending a cyclic vector e⊗m
to rm
θ
Proof . A relation in the fusion product means that some expression of the form
X
αi1 ...is x1 ⊗ ti1 · · · xs ⊗ tis
(26)
with fixed t degree can be expressed in g⊗m via a linear combination of monomials of lower
t-degree. The coefficients of the expression of (26) in terms of the lower degree monomials are
polynomials in evaluation parameters c1 , . . . , cn . Therefore, by the very definition of the action
of Pm on Wm , the operator of the form (26), which is a relation in the fusion product, vanishes
on (Wm /Pm+ Wm )∗ .
sym
Note that Wm and Vm are naturally graded by the degree grading on Pm (Pm
). This grading
sym
defines a graded character of the space Vm /Pm+ .
2
Corollary 5. chq Vm /Pm+ Vm ≤ q m chq−1 g∗∗m .
Proof . Follows from Proposition 5 and Proposition 3. Note that the factor m2 is a degree of
0 .
the cyclic vector rm
Remark 7. In the next section we combine Corollary 5 with Corollary 4 in order to compute
the character of Lgr .
6
Proofs of the main statements
2
Proposition 6. chq (Lm )∗ ≤ chq Vm ≤ q m chq−1 g∗∗m /(q)m .
Proof . We know that
chq L∗m ≤ ch(L0m )∗ = chq Vm .
2
Because of the surjection (25), the character of Vm∗ is smaller than or equal to q m chq−1 g∗∗m /(q)m
(since 1/(q)m is the character of the space of symmetric polynomials in m variables). This proves
the Proposition.
Theorem 4. Lm ' L0m .
Proof . Corollary 4 provides an inequality
2
chq Lm
qm
ch −1 g∗∗m .
≥
(q)m q
Now from Proposition 6 we obtain
2
2
qm
qm
chq−1 g∗∗m ≤ chq Lm ≤ chq L0m ≤
ch −1 g∗∗m .
(q)m
(q)m q
Theorem is proved.
18
E. Feigin
sym
Corollary 6. The dual module (Vm /Pm+
Vm )∗ and g∗∗m are isomorphic as g ⊗ C[t] modules.
Proof . Follows from Propositions 5, 3 and Theorem 4.
Corollary 7. We have an isomorphism of gab ⊗ C[t−1 ] modules
Lgr ' U gab ⊗ C[t−1 ] /I,
where I is the minimal g invariant ideal containing the coefficients of the current eθ (z)2 = 0.
Remark 8. Corollary 7 is a generalization of the sl2 case from [13]. It also proves a level 1
conjecture from [10].
sym
Corollary 8. The action of the polynomial ring Pm
on Vm is free.
sym ∗
Proof . Follows from the isomorphism (Lm /Pm+
) ' g∗∗m and the character equality
2
chq Lm =
qm
2
sym
chq−1 g∗∗m = q m chq Pm
· chq−1 g∗∗m .
(q)m
Recall the vectors
wi1 ,...,im = eθ ⊗ ti1 · · · eθ ⊗ tim v0 ∈ L.
Let w̄i1 ,...,im ∈ Lm be the images of these vectors.
Corollary 9. Lm is generated by the set of vectors
w̄i1 ,...,im ,
−m ≥ i1 ≥ · · · ≥ im
(27)
with the action of the algebra U(g ⊗ C[t]).
Proof . Recall the element
Y
r m = z1 · · · zm
∈ Vm .
(zi − zj )2 ⊗ e⊗N
θ
1≤i<j≤m
sym
Since rm is a cocyclic vector of Vm /Pm+
and the polynomial algebra acts freely on Vm , we
sym
obtain that the space Pm+
rm is a cocyclic subspace of Vm , which means that for any vector
sym
sym
v ∈ Vm the space U(g ⊗ C[t]) · v has a nontrivial intersection with Pm+
rm . We note that Pm+
rm
coincides with the subspace of Vm of g weight mθ. Dualizing this construction we obtain that
the subspace of Lm of g weight −mθ is U(g ⊗ C[t]) cyclic. This space is linearly spanned by the
set (27). Corollary is proved.
Corollary 10. The space grm F• (N ) is generated by the set of vectors
w̄i1 ,...,im ,
−m ≥ i1 ≥ · · · ≥ im ≥ −N
(28)
with the action of the algebra U(g ⊗ C[t]).
Proof . Introduce an increasing filtration J• on Lm :
Jn = U(g ⊗ C[t]) · span w̄i1 ,...,im : −i1 − · · · − im ≤ m2 + n .
Corollary 6 provides an isomorphism of g ⊗ C[t] modules
∗∗m
U(g ⊗ C[t]) · w̄i1 ,...,im / (J−i1 −···−im −1 ∩ U(g ⊗ C[t]) · w̄i1 ,...,im ) ' g
.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
19
Thus, since all vectors of the form (28) belong to Fm (N ), we obtain
N
dim grm F• (N ) ≥
(dim g)m .
m
Because of the equality
N X
N
dim D(N ) = (1 + dim g)N =
m=0
we conclude that dim grm F• (N ) =
rated by the vectors (28).
N
m
m
(dim g)m ,
(dim g)m and hence the whole space grm F• (N ) is gene
Proposition 7. The induced PBW filtration F• (N ) ,→ D(N ) coincides with the tN -filtration G• ,
i.e. Fm (N ) = GN −m .
Proof . Recall (see Lemma 4) that GN −m ,→ Fm (N ). Since v0 is proportional to (fθ ⊗ tN )N vN ,
we obtain that w̄i1 ,...,im ∈ GN −m for −m ≥ i1 ≥ · · · ≥ im ≥ −N . Therefore, Corollary 10 gives
Fm (N ) ,→ GN −m . Proposition is proved.
∗∗m . The character
Corollary 11. The graded component grm F• (N ) is filtered by N
m copies of g
N
2
.
of the space of cyclic vectors of those fusions is equal to q m
m q
We summarize all above in the following theorem:
Theorem 5. Let F• be the PBW filtration on the level one vacuum b
g module L. Then
a) grm F• is filtered by the fusion modules g∗∗m .
b) The character of the space of cyclic vectors of these g∗∗m is equal to
2
qm
(q)m .
c) The induced PBW filtration on Demazure modules D(N ) coincides with the double fusion
filtration coming from Theorem 3.
d) The defining relation in Lgr is eθ (z)2 = 0.
A list of the main notations
g – simple finite-dimensional algebra;
b
g – corresponding affine Kac–Moody algebra;
θ – highest weight of the adjoint representation of g;
eθ , fθ ∈ g – highest and lowest weight vectors of the adjoint representation;
L – the basic (vacuum level one) representation of b
g;
v0 ∈ L – a highest weight vector of L;
vN ∈ L – an extremal vector of the weight N θ;
D(N ) ,→ L – Demazure module with a cyclic vector vN ;
grm A• – m-th graded component of the associated graded space with respect to the
filtration A• ;
F• – (increasing) PBW filtration on L;
20
E. Feigin
F• (N ) = F• ∩ D(N ) – induced PBW filtration on D(N );
G• – (decreasing) tN -filtration on D(N );
V1 ∗ · · · ∗ VN – associated graded g ⊗ C[t] module with respect to the fusion filtration on
the tensor product of cyclic g ⊗ C[t] modules;
V1 ∗ ∗ · · · ∗ ∗VN – associated graded g ⊗ C[t] module with respect to the fusion filtration
on the tensor product of cyclic g modules;
grm (V1 ∗ · · · ∗ VN ) – m-th graded component with respect to the fusion filtration;
chq – a graded character defined by the operator d;
¯
chq – a graded character defined by the operator d.
Acknowledgements
EF thanks B. Feigin and P. Littelmann for useful discussions. This work was partially supported
by the RFBR Grants 06-01-00037, 07-02-00799 and NSh-3472.2008.2, by Pierre Deligne fund
based on his 2004 Balzan prize in mathematics, by Euler foundation and by Alexander von
Humboldt Fellowship.
References
[1] Ardonne E., Kedem R., Fusion products of Kirillov–Reshetikhin modules and fermionic multiplicity formulas,
J. Algebra 308 (2007), 270–294, math.RT/0602177.
[2] Ardonne E., Kedem R., Stone M., Fusion products, Kostka polynomials and fermionic characters of sc
u(r +
1)k , J. Phys. A: Math. Gen. 38 (2005), 9183–9205, math-ph/0506071.
[3] Frenkel E., Ben-Zvi D., Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, Vol. 88,
Amer. Math. Soc., Providence, RI, 2001.
[4] Chari V., Le T., Representations of double affine Lie algebras, in A Tribute to C.S. Seshadri (Chennai,
2002), Trends Math., Birkhäuser, Basel, 2003, 199–219, math.QA/0205312.
[5] Chari V., Loktev S., Weyl, Demazure and fusion modules for the current algebra of slr+1 , Adv. Math. 207
(2006), 928–960, math.QA/0502165.
[6] Chari V., Presley A., Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001),
191–223, math.QA/0004174.
[7] Demazure M., Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7
(1974), 53–88.
[8] Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Springer-Verlag, New York, 1997.
b 2 cosets, J. Lie Theory 17 (2007), 145–161, math.QA/0603226.
[9] Feigin E., Infinite fusion products and sl
[10] Feigin E., The PBW filtration, math.QA/0702797.
[11] Feigin B., Feigin E., Q-characters of the tensor products in sl2 case, Moscow Math. J. 2 (2002), 567–588,
math.QA/0201111.
c2 -modules as infinite tensor products, in Fundamental Mathematics Today,
[12] Feigin B., Feigin E., Integrable sl
Editors S. Lando and O. Sheinman, Independent University of Moscow, 2003, 304–334, math.QA/0205281.
[13] Feigin B., Feigin E., Jimbo M., Miwa T., Takeyama Y., A φ1,3 -filtration on the Virasoro minimal series
M (p, p0 ) with 1 < p0 /p < 2, Publ. Res. Inst. Math. Sci. 44 (2008), 213–257, math.QA/0603070.
[14] Feigin B., Jimbo M., Kedem R., Loktev S., Miwa T., Spaces of coinvariants and fusion product. I. From
equivalence theorem to Kostka polynomials, Duke. Math. J. 125 (2004), 549–588, math.QA/0205324.
c2
Feigin B., Jimbo M., Kedem R., Loktev S., Miwa T., Spaces of coinvariants and fusion product. II. sl
character formulas in terms of Kostka polynomials, J. Algebra 279 (2004), 147–179, math.QA/0208156.
[15] Frenkel I.B., Kac V.G., Basic representations of affine Lie algebras and dual resonance models, Invent.
Math. 62 (1980/1981), 23–66.
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
21
[16] Feigin B., Kirillov A.N., Loktev S., Combinatorics and geometry of higher level Weyl modules, in Moscow
Seminar on Mathematical Physics. II, Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc., Providence, RI, 2007, 33–47, math.QA/0503315.
[17] Feigin B., Loktev S., On generalized Kostka polynomials and quantum Verlinde rule, in Differential Topology,
Infinite-Dimensional Lie Algebras and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math.
Soc., Providence, RI, 1999, 61–79, math.QA/9812093.
[18] Fourier G., Littelmann P., Tensor product structure of affine Demazure modules and limit constructions,
Nagoya Math. J. 182 (2006), 171–198, math.RT/0412432.
[19] Fourier G., Littelmann P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and
limit constructions, Adv. Math. 211 (2007), 566–593, math.RT/0509276.
[20] Hernandez D., Quantum toroidal algebras and their representations, Selecta Math., to appear,
arXiv:0801.2397.
[21] Kac V., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
[22] Kac V., Vertex algebras for begginers, University Lecture Series, Vol. 10. Amer. Math. Soc., Providence,
RI, 1997.
[23] Kedem R., Fusion products, cohomology of GLN flag manifolds, and Kostka polynomials, Int. Math. Res.
Not. 2004 (2004), no. 25, 1273–1298, math.RT/0312478.
[24] Kreiman V., Lakshmibai V., Magyar P., Weyman J., Standard bases for affine SL(n)-modules, Int. Math.
Res. Not. 2005 (2005), no. 21, 1251–1276, math.RT/0411017.
[25] Kumar S., Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics,
Vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002.
[26] Loktev S., Weight multiplicity polynomials of multi-variable Weyl modules, arXiv:0806.0170.